A375912 Primes p such that p*nextprime(p)+1 and p + nextprime(p)+1 are both perfect squares where nextprime(p) is the smallest prime that is larger than p.

3, 11, 59, 179, 311, 419, 2111, 3119, 5099, 21011, 21839, 24419, 30011, 37811, 41759, 44699, 60899, 68819, 83639, 86111, 100799, 135719, 143111, 161879, 163019, 165311, 177011, 210599, 218459, 241511, 273059, 304979, 312839, 437111, 450299, 491039, 584279, 595139, 603899, 637319

Offset: 1

Author

Michel Lagneau, Sep 02 2024

Keywords

• nonn

Examples

			11 is a term because 11*nextprime(11)+1 = 12^2 and 11 + nextprime(11)+1 = 5^2.
		

Crossrefs

Intersection of A001359 and A242384.

Programs

Maple

nn:=10^5:
for n from 1 to nn do:
p:=ithprime(n):q:=nextprime(p):p1:=sqrt(p*q+1):p2:=sqrt(q+p+1):
 if floor(p1) = p1 and floor(p2)=p2
  then
  printf(`%d, `,p):
  else
 fi:
od:

Mathematica

Select[Partition[Prime[Range[100000]], 2, 1], IntegerQ[Sqrt[#[[1]] + #[[2]] + 1]] && IntegerQ[Sqrt[#[[1]]*#[[2]] + 1]] &][[;; , 1]] (* Amiram Eldar, Sep 02 2024 *)